The heuristic for the oblong cubic surface

Let $k>1$ be a square-free integer, and let $U\subset\mathbb{A}^3$ be the cubic surface

$$x^3+ky^3+kz^3=1.$$

Let $D(\mathbb{R})$ denote the real component appearing in the paper's heuristic, and let $U(\mathbf{A}^{\mathrm{fin}}_{\mathbb{Z}})$ denote the finite integral adelic points. Then $\gamma_U$ is the exponent in the paper's main heuristic.

Oblong-surface conjecture. The main heuristic holds with

$$\gamma_U=\frac{3}{8}$$

and

$$V=D(\mathbb{R})\times U(\mathbf{A}^{\mathrm{fin}}_{\mathbb{Z}}).$$

The source presents this as a concrete test of the heuristic; it notes that $b=1$ and $\varrho_U=2$ for this surface, but supplies no resolution status.